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Mirrors > Home > MPE Home > Th. List > spw | Structured version Visualization version GIF version |
Description: Weak version of the specialization scheme sp 2226. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2226 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2226 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2188 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2226 are spfw 2142 (minimal distinct variable requirements), spnfw 2106 (when 𝑥 is not free in ¬ 𝜑), spvw 2088 (when 𝑥 does not appear in 𝜑), sptruw 1907 (when 𝜑 is true), and spfalw 2107 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.) |
Ref | Expression |
---|---|
spw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
spw | ⊢ (∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 2011 | . 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓) | |
2 | ax-5 2011 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
3 | ax-5 2011 | . 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑) | |
4 | spw.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 1, 2, 3, 4 | spfw 2142 | 1 ⊢ (∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∀wal 1656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1881 |
This theorem is referenced by: hba1w 2151 spaev 2154 ax12w 2186 bj-ssblem1 33168 bj-ax12w 33201 |
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